Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X, \tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb {F}_n^+$. They develop the necessary dilation theory for both models.